Diffraction by an Infinite Slit

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Abstract
The diffraction of plane waves by an infinite slit is investigated, with attention drawn to the case of grazing incidence and for wavelengths short compared to the slit width. The wave pattern is time harmonic and two dimensional, with identical behavior in all planes normal to the slit axis. At the coplanar screens bordering the slit, the normal derivative of the wave function is assumed to vanish, for this boundary condition provides a problem with calculable diffraction even at grazing incidence. A useful formulation (Sec. 2) of the boundary value problem involves Fourier transforms of field distributions in the plane of the screens, and enables the transmission cross section of the slit to be directly inferred (Sec. 3). The screen distributions are characterized by a pair of integral equations which allow systematic approximation (Sec. 4) at short wavelengths for arbitrary angle of incidence. A few terms in the asymptotic development of the cross section at oblique incidence are obtained explicitly, and since this development fails at grazing incidence, the analogous terms for the latter case are derived by a limiting process. Lastly (Sec. 5), in the related problem of planewave scattering by an infinite strip, a comparison is made with the variational results based on strip distributions of primary or unperturbed form.

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